When 900 is divided by positive integer d, the remainder is

[BTGmoderatorDC]

When 900 is divided by positive integer d, the remainder is r. For some integer N > 5000, when N is divided by positive integer D, the remainder is R. Is R > d?

(1) r = 1

(2) D = 23

OA C

Source: Magoosh

[fskilnik@GMATH]

When 900 is divided by positive integer d, the remainder is r. For some integer N > 5000, when N is divided by positive integer D, the remainder is R. Is R > d?

(1) r = 1

(2) D = 23
Source: Magoosh

Very nice conceptual problem!
$$d,D\,\,\, \ge 1\,\,\,;\,\,\,\,N\,\,\, \ge \,\,5001\,\,\,\,\,\,\left( {\,{\rm{ints}}\,} \right)$$
$$\left\{ \matrix{
900 = {Q_1} \cdot d + r\,\,\,\,\,\,\left[ {\,0 \le r < d\,\,,\,\,\,\,r\,\,{\mathop{\rm int}} \,} \right]\,\,\,\,\,\,\,\left( * \right) \hfill \cr
N = {Q_2} \cdot D + R\,\,\,\,\,\,\,\left[ {\,0 \le R < D\,\,,\,\,\,R\,\,{\mathop{\rm int}} \,} \right]\,\,\,\,\,\left( {**} \right) \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,\,\,\,\,R\,\,\mathop > \limits^? \,\,d$$

The BIFURCATION of each statement alone will be omitted because it is easy (also intuitively).
(It is important, though. If you feel you need to do it, do not hesitate and hands-on!)

$$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{
\,r = 1\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,{Q_1} \cdot d\, = 899 = 29 \cdot 31\,\,\,\left( {{\rm{primes}}} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,d = \,\,29\,,\,31\,\,{\rm{or}}\,\,899\,\,\,\left( {d > r = 1} \right)\,\,\,\,\, \hfill \cr
D = 23\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,0 \le R < 23\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,R\,\,\mathop > \limits^? \,\,d\,\,\, \ldots \,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

P.S.: Jay´s approach to factorize 899 is excellent. Do not miss it (post below)!

[Jay@ManhattanReview]

When 900 is divided by positive integer d, the remainder is r. For some integer N > 5000, when N is divided by positive integer D, the remainder is R. Is R > d?

(1) r = 1

(2) D = 23

OA C

Source: Magoosh

Let's take each statement one by one.

(1) r = 1

Given that when 900 is divided by positive integer d, the remainder is r = 1, we can conclude that (900 - 1) = 899 is a factor of d. Since we do not know anything about D and R, we cannot decide. Insufficient.

(2) D = 23

=> R < D => R < 23; since Remainder < Divisor

Since we do not know anything about d and r, we cannot decide. Insufficient.

(1) and (2) together

From Statement 1, we know that d is a factor of 899. Let's find out the factors of 899, which can be time-consuming. Since 899 is 1 away from 900, we take the route of a^2 - b^2.

899 = 900 - 1 = 30^2 - 1^2 = (30 + 1)*(30 - 1) = 31*29

Since both 31 and 29 are prime, we have factors of 899 as {1, 29, 31, 899}

d cannot be 1 since if d = 1, r (= remainder) cannot be 1. r < d.

So, d can be 29, 31, or 899.

From Statement 2, we know that the maximum value of R is 22 and we just deduced that the minimum value of d = 29, so, R < d. The answer to the question is definitely "No." Sufficient.

The correct answer: C

Hope this helps!

-Jay
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[Scott@TargetTestPrep]

When 900 is divided by positive integer d, the remainder is r. For some integer N > 5000, when N is divided by positive integer D, the remainder is R. Is R > d?

(1) r = 1

(2) D = 23

OA C

Solution:


Statement One Alone:
r = 1

Since the remainder when 900 is divided by d is 1, we can write 900 = dk + 1 for some integer k. Then, dk = 899. Since 899 = 29 x 31, it follows that d can be 29, 31 or 899.

If N = 5001 and D = 4101, then R = 900. In this case, R > d for any possible value of d. However, if N = 5001 and D = 3, then R = 0 since 5001 is divisible by 3. In this scenario, R < d. Since we have multiple answers to the question depending on the values of the variables, statement one alone is not sufficient to determine the answer to the question.

Eliminate answer choices A and D.

Statement Two Alone:
D = 23

Since N can be any integer greater than 5000 and since D = 23, R can be any integer between 0 and 23, including 0. If d = 24, then d > R and the answer to the question is “No!”. On the other hand, if d = 1 and R = 2 (for instance, if N = 5016), then R > d. In this case, the answer to the question is “Yes!”. Since we have multiple answers to the question depending on the values of the variables, statement two alone is not sufficient to determine the answer to the question.

Eliminate answer choice B.

Statements One and Two Together:

Using statement one, we know d can only be 29, 31 or 899. Using statement two, we know R < 23. Thus, d is greater than R. The answer to the question “Is R > d?” is “No!”. Statements one and two together are sufficient to answer the question.

Answer: C